reserve n,m for Element of NAT;
reserve h,k,r,r1,r2,x,x0,x1,x2,x3 for Real;
reserve f,f1,f2 for Function of REAL,REAL;
reserve S for Seq_Sequence;

theorem Th34:
  (cdif(f1(#)f2,h).1).x
     = f1.(x+h/2)* (cdif(f2,h).1).x + f2.(x-h/2)* (cdif(f1,h).1).x
proof
  (cdif(f1(#)f2,h).1).x = (cdif(f1(#)f2,h).(0+1)).x
    .= cD(cdif(f1(#)f2,h).0,h).x by DIFF_1:def 8
    .= cD(f1(#)f2,h).x by DIFF_1:def 8
    .= (f1(#)f2).(x+h/2) - (f1(#)f2).(x-h/2) by DIFF_1:5
    .= f1.(x+h/2)* f2.(x+h/2) - (f1(#)f2).(x-h/2) by VALUED_1:5
    .= f1.(x+h/2)* f2.(x+h/2) - f1.(x-h/2)* f2.(x-h/2) by VALUED_1:5
    .= f1.(x+h/2)* (f2.(x+h/2)-f2.(x-h/2))
       + f2.(x-h/2)* (f1.(x+h/2)-f1.(x-h/2))
    .= f1.(x+h/2)* cD(f2,h).x + f2.(x-h/2)* (f1.(x+h/2)-f1.(x-h/2)) by DIFF_1:5
    .= f1.(x+h/2)* cD(f2,h).x + f2.(x-h/2)* cD(f1,h).x by DIFF_1:5
    .= f1.(x+h/2)* cD(cdif(f2,h).0,h).x
       + f2.(x-h/2)* cD(f1,h).x by DIFF_1:def 8
    .= f1.(x+h/2)* cD(cdif(f2,h).0,h).x
       + f2.(x-h/2)* cD(cdif(f1,h).0,h).x by DIFF_1:def 8
    .= f1.(x+h/2)* cdif(f2,h).(0+1).x
       + f2.(x-h/2)* cD(cdif(f1,h).0,h).x by DIFF_1:def 8
    .= f1.(x+h/2)* (cdif(f2,h).1).x + f2.(x-h/2)* (cdif(f1,h).1).x
                                                             by DIFF_1:def 8;
  hence thesis;
end;
